Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

Abstract

The goal of this paper is to establish the applicability of the Lyapunov–Schmidt reduction and the Centre Manifold Theorem (CMT) for a class of hyperbolic partial differential equation models with nonlocal interaction terms describing the aggregation dynamics of animals/cells in a one-dimensional domain with periodic boundary conditions. We show the Fredholm property for the linear operator obtained at a steady-state and from this establish the validity of Lyapunov–Schmidt reduction for steady-state bifurcations, Hopf bifurcations and mode interactions of steady-state and Hopf. Next, we show that the hypotheses of the CMT of Vanderbauwhede and Iooss (Center manifold theory in infinite dimensions. In: Jones, C., Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1, pp. 125–163. Springer, Berlin, 1992) hold for any type of local bifurcation near steady-state solutions with SO(2) and O(2) symmetry. To put our results in context, we review applications of hyperbolic partial differential equation models in physics and in biology. Moreover, we also survey recent results on Fredholm properties and Centre Manifold reduction for hyperbolic partial differential equations and equations with nonlocal terms.